Geometry of rare regions behind Griffiths singularities in random quantum magnets

TL;DR

This paper investigates the geometrical structure of rare regions responsible for Griffiths singularities in disordered quantum magnets, revealing their non-compact, fractal-like shapes across different models and dimensions.

Contribution

It provides a detailed analysis of the geometry of rare regions in disordered quantum systems, highlighting their non-compact, fractal nature and differences between diluted and random models.

Findings

Rare regions grow logarithmically with system size.

In diluted models, rare regions are isotropic and tree-like.

In random models, rare regions are quasi-one-dimensional.

Abstract

In many-body systems with quenched disorder, dynamical observables can be singular not only at the critical point, but in an extended region of the paramagnetic phase as well. These Griffiths singularities are due to rare regions, which are locally in the ordered phase and contribute to a large susceptibility. Here, we study the geometrical properties of rare regions in the transverse Ising model with dilution or with random couplings and transverse fields. In diluted models, the rare regions are percolation clusters, while in random models the ground state consists of a set of spin clusters, which are calculated by the strong disorder renormalization method. We consider the so called energy cluster, which has the smallest excitation energy and calculate its mass and linear extension in one-, two- and three-dimensions. Both average quantities are found to grow logarithmically with the linear size of the sample. Consequently, the rare regions are not compact: for the diluted model they are isotropic and tree-like, while for the random model they are quasi-one-dimensional.